cayley-Hamilton law

1)
there is
A.we define by for every and in
does f define inner product in ?
how i tried:
in the book by definition inner product is following these rules:
(u,v)=(v,u)
(u+v,w)=(u,w)+(v,w)
(ku,v)=k(u,v)
i dont know how to apply these rules here
?
B.find a simetric matrices B so
(clue: use cayley-Hamilton law)
?
how i tried:
hamilton law say that the representative matrice turns to zero the minimal polinomial M(A)=0
any guidence
?